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  2. Pythagorean triple - Wikipedia

    en.wikipedia.org/wiki/Pythagorean_triple

    Every Pythagorean triple is a Heronian triple, because at least one of the legs a, b must be even in a Pythagorean triple, so the area ab/2 is an integer. Not every Heronian triple is a Pythagorean triple, however, as the example (4, 13, 15) with area 24 shows.

  3. Tree of primitive Pythagorean triples - Wikipedia

    en.wikipedia.org/wiki/Tree_of_primitive...

    A Pythagorean triple is a set of three positive integers a, ... but each set separately produces all primitive triples. For example, using [5, 12, 13] as the parent ...

  4. Formulas for generating Pythagorean triples - Wikipedia

    en.wikipedia.org/wiki/Formulas_for_generating...

    Wade and Wade [17] first introduced the categorization of Pythagorean triples by their height, defined as c − b, linking 3,4,5 to 5,12,13 and 7,24,25 and so on. McCullough and Wade [18] extended this approach, which produces all Pythagorean triples when k > h √ 2 /d: Write a positive integer h as pq 2 with p square-free and q positive.

  5. Pythagorean theorem - Wikipedia

    en.wikipedia.org/wiki/Pythagorean_theorem

    In other words, a Pythagorean triple represents the lengths of the sides of a right triangle where all three sides have integer lengths. [1] Such a triple is commonly written (a, b, c). Some well-known examples are (3, 4, 5) and (5, 12, 13). A primitive Pythagorean triple is one in which a, b and c are coprime (the greatest common divisor of a ...

  6. Plimpton 322 - Wikipedia

    en.wikipedia.org/wiki/Plimpton_322

    This table lists two of the three numbers in what are now called Pythagorean triples, i.e., integers a, b, and c satisfying a 2 + b 2 = c 2. From a modern perspective, a method for constructing such triples is a significant early achievement, known long before the Greek and Indian mathematicians discovered solutions to this problem.

  7. Group of rational points on the unit circle - Wikipedia

    en.wikipedia.org/wiki/Group_of_rational_points...

    3/5 + 4/5i and 5/13 + 12/13i (which correspond to the two most famous Pythagorean triples (3,4,5) and (5,12,13)) are rational points on the unit circle in the complex plane, and thus are elements of G. Their group product is −33/65 + 56/65i, which corresponds to the Pythagorean triple (33,56,65). The sum of the squares of the numerators 33 ...

  8. Pythagoreanism - Wikipedia

    en.wikipedia.org/wiki/Pythagoreanism

    The Pythagorean schools and societies died out from the 4th century BC. Pythagorean philosophers continued to practice, albeit no organised communities were established. [14] According to surviving sources by the Neopythagorean philosopher Nicomachus, Philolaus was the successor of Pythagoras. [16] According to Cicero (de Orat.

  9. Pell number - Wikipedia

    en.wikipedia.org/wiki/Pell_number

    If a right triangle has integer side lengths a, b, c (necessarily satisfying the Pythagorean theorem a 2 + b 2 = c 2), then (a,b,c) is known as a Pythagorean triple. As Martin (1875) describes, the Pell numbers can be used to form Pythagorean triples in which a and b are one unit apart, corresponding to right triangles that are nearly isosceles ...